A Multivariate Interlace Polynomial and its Computation for Graphs of Bounded Clique-Width

Bruno Courcelle

Abstract

We define a multivariate polynomial that generalizes in a unified way the two-variable interlace polynomial defined by Arratia, Bollobás and Sorkin on the one hand, and a one-variable variant of it defined by Aigner and van der Holst on the other.

We determine a recursive definition for our polynomial that is based on local complementation and pivoting like the recursive definitions of Tutte's polynomial and of its multivariate generalizations are based on edge deletions and contractions. We also show that bounded portions of our polynomial can be evaluated in polynomial time for graphs of bounded clique-width. Our proof uses an expression of the interlace polynomial in monadic second-order logic, and works actually for every polynomial expressed in monadic second-order logic in a similar way.